Category Archives: STEM

A few years ago, I switched one of my first year courses to use what I call a semi-open-book approach.

Open-book exams of course allow students to bring whatever materials they wish into them, but they have the disadvantage that students will often bring in materials that they have not studied in detail, or even at all. In such cases, sifting through materials to help them answer a question could be counter productive.

On the other hand, the real world is now an increasingly “open-book” environment, which huge amounts of information available to those in the workplace which is now almost always Internet connected.

So I decided to look at another approach. Students are allowed to bring in a single, personalised, A4 sheet, on which they can write whatever they wish on both sides. There are a few rules:

the sheet must be written on “by hand”, that is to say, it cannot be printed to from a computer, or typed;

the sheet must be “original”, that is to say, it cannot be a photocopy of another sheet (though students may of course copy their original for reference);

the sheet must be the student’s own work, and they must formally declare as much (with a tick box);

the sheet must be handed in with the exam paper, although it is not marked.

The purpose of these restrictions are to ensure that each student takes a lead in producing an individual sheet, and to inhibit cottage industries of copied sheets.

In terms of what can go on the sheet? Well anything really. It can be sections from notes, important formulae, sample questions or solutions. The main purpose here is to prompt students to work out what they would individually distill down to an A4 page. So they go through all the module notes, tutorial problems and more, and work out the most valuable material that deserves to go on one A4 page. I believe that this process itself is the greatest value of the sheet, its production rather than its existence in the exam. I’m working on some research to test this.

So I email them each an A4 PDF, which they can print out at home, and on whatever colour paper they may desire. The sheet is individual and has their student number on it with a barcode, for automated processing and analysis afterwards for a project I’m working on, but this is anonymised. The student’s name in particular does not appear, since in Ulster University, it does not appear on the exam booklet.

The top of my sheet looks like this:

The top of a sample guide sheet.

So, if you would like to do the same, I am enclosing the Python script, and LaTeX that I use to achieve this. You could of course use any other technology, or not individualise the sheet at all.

For convenience the most recent code will also be placed on a GitHub repository here, feel free to clone away.

My script has just been rewritten for Python 3.x, and I’ve added a lot of command line parameters to decouple it from me and Ulster University only use. It opens a CSV file from my University which contains student id numbers, student names, and emails in specific columns. These are the default for the script but can be changed. For each student it uses LaTeXÂ to generate the page. It actually creates inserts for each student of the name and student number, you can then edit open-book.tex to allow the page to be as you wish it. You don’t need to know much LaTeX to achieve this, but ping me if you need help. I am also using a LaTeX package to create the barcodes automatically.

I’ve spent a bit of time adding command line parameters to this script, but you can try using

1

python3 open-book.py--help

for information. The script has been rewritten for Python 3. If you run it without parameters it will enter interactive mode and prompt you.

I’d strongly recommend running with the –test-only option at first to make sure all looks good, and opening open-book.pdf will show you the last generated page so you can see it’s what you want.

Anyway, feel free to do your own thing, or mutilate the code. Enjoy!

Python script to email students from a CSV file

Python

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#!/usr/bin/env python

#

# Copyright Colin Turner 2014-2017

#

# Free and Open Source Software under GPL v3

#

importargparse

importcsv

importre

importsubprocess

importsmtplib

fromemail.mime.application importMIMEApplication

fromemail.mime.multipart importMIMEMultipart

fromemail.mime.text importMIMEText

defprocess_student(args,row):

"""Takes a line from the CSV file, you will likely need to edit aspects of this."""

'kicks of the processing of a single student'

student_number=row[args.student_id_column]

student_name=row[args.student_name_column]

student_email=row[args.student_email_column]

print(' Processing:',student_name,':',student_email)

create_latex_inserts(student_number,student_name)

create_pdf()

send_email(args,student_name,student_email)

defcreate_latex_inserts(student_number,student_name):

"""Write LaTeX inserts for the barcode and student name

For each student this will create two tiny LaTeX files:

* open-book-insert-barcode.tex which contains the LaTeX code for a barcode representing the student number

* open-book-insert-name.tex which will contain simply the student's name

These files can be included/inputted from open-book.tex as desired to personalise that document

student_number is the ID in the students record system for the student

A friend of mine recently posted about a chain of events – people sharing birthdays – that was so unlikely that a lottery ticket purchase was called for. Most people might make similar comments as the oddity of these events struck them. There followed some discussion about these problems and it made me think of trying to set out the issues more carefully.

I’ve done a lecture, in some variations (for school audiences), almost every year about the deceptiveness of common sense in mathematics. Indeed, it provided the title of this blog.

The Â Birthday Problem

One of the problems I examined is the famous birthday problem. Let’s look at two aspects of this.

How many people must be in a room before we areÂ certain that two of them share the same birthday?

This can be answered by using the so called pigeon hole principle. In this case, the pigeon holes are all the days of the calendar in a year, 365, or let’s say 366 to allow for the worst case scenario of a leap year. The people in the room are the pigeons. You can place 366 pigeons into the holes and keep them all separate, but any more ensures that at least two must share one of the holes. So, similarly, you need 367 people to be absolutely sure that one pair in the room shares a birthday.

Let’s look at a more subtle question.

How many people must be in a room so that it is more likely than not that two of them share the same birthday?

It turns out this number is deceptively low. We reach a 50% chance (more likely than not) with just 23 people in the room. Perhaps even more surprisingly we reach a 99.9% chance with 70 people in the room. The calculations for the probabilities are set out in the link to the lecture, and the wikipedia articleÂ as mentioned above, but I don’t want to focus on that. I want to focus on why the result is so shocking to most people.

The issue is the collision of relatively rare events (a birthday) with numerous combinations. For instance if you want to consider how many pairs one can form in the room we can see this.

If there are three people in the room, A, B and C, there are three possible pairs: AB, AC, and BC. If there are four people in the room, A, B, C and D, there are six possible pairs: AB, AC, AD, BC, BD, and CD. There’s actually a nice formula for this and you can work this out for more examples.

You don’t need to know the formula, but if you want it isÂ

People in Room

Possible Pairs

3

3

4

6

10

45

15

105

20

190

23

253

30

435

50

1225

70

2415

You can see just how rapidly the number of pairs increases. It might seem un-obvious that 23 people is as likely as not to produce a duplicate birthday, but it’s perhaps less so when you realise that we are not predicting which pigeon hole has two pigeons in it to go back to our analogy. But given 253 possible pairings, it seems less unlikely even intuitively that there will be “collisions” within one of those pairs.

Coin Tosses

Let me talk about a similar problem, flipping a (fair) coin to get a Head (H) or a Tail (T). Let’s consider that we flip the coin three times. The possible outcomes are:

HHH,Â HHT,Â HTH,Â THH,Â TTH,Â THT,Â TTH, TTT.

These are all absolutely equally likely, and even this is sometime unintuitive. The coin has no memory of how it performed, and no desire to conform to some concept such as a “law of averages“. But we tend to categorise these 8 possible events into 4 categories.

Three Heads: (one way this can happen, so 1/8 probability)

Two Heads, One Tail: (three ways this can happen, so 3/8 probability)

One Head, Two Tails: (three ways this can happen, so 3/8 probability)

Three Tails: (one way this can happen so 1/8 probability).

In doing so, we can see why the HHH and TTT events look particularly exciting and rare. They are no more rare than any otherÂ individual outcome, but it’s the groupings of how one can get two heads that makes that collection of events more unexciting.

To take this to more of an extreme. If we flip a coin ten times, then these outcomes are all exactly equally likely.

HHHHHHHHHH

HTHTHTHTHT

HHTHTTHTTH

THTTHHHTHT

TTTTTTTTTT

But some will be far more remarked upon that others. But even if we have an experiment that flips a coin 10 times over and over again enough times, we will expect to see all of these (more or less, eventually) equally often, but we will probably not consider these exciting except maybe outcomes 1,5 and maybe 2.

Genealogy

Another example, closer to home. Let’s assume that there have been 7,500 generations of Homo Sapiens since its emergence as a species.

The probability that in each of those 7,500 generations in both your paternal and maternal line that all the pairings took place between the right people at the right time in the right circumstances to give rise toÂ you is astronomical. You have two parents, four grand parents, 8 great grand parents and so on. Just for that collection of people. If we add you into the mix, then the number of pairings involved over these 7500 generations is

That’s the number of people involved to get to you, including you. They all had to survive and meet at the right time, and um, do everything else at exactly the right time in the right way. What are the odds? And yet you are here… You might find that exceptional, and of course in a way it is, but the point is, you are not more likely or unlikely than most other human beings that have been born, or countless potential humans who might have been but were not born because their parents never even met. The fact that you specifically exist is unlikely, but the fact that if you did not someone else would exist instead is highly likely and predictable.

Naturally, however, we tend to think of our own existence rather more than some random person on the other side of the planet. But each one of us is an example of the coin flipping perfectly as needed every single time. Sometimes a Head, sometimes a Tail, but that exact perfect sequence essential to produce you. Naturally we are each quite taken with the outcome that produces us and also our nearest and dearest. But if it had not happened, we wouldn’t be here to wonder about it.

Availability and Psychology

As you can see there is more to this than mathematics and probability, there is also perception and psychology, specifically a concept called the Availability Heuristic. Some things stand out to us as being significant when they are no more significant than other events we ignore.

You might live in a city of a million people. You might personally know only a hundred. You might pass a thousand on your commute to work each day. You do that hundreds of times a year. The number of possible combinations of people you will see over a year is enormous, but you won’t remember it as a special event when you bump into all the people who aren’t dear to you.

The Anthropic Principle

A nice example of this thinking is the question that many people have asked which is essentially: why is the Universe, and our Earth, set up so nicely to allow humans to exist? If certain physical constants were just a bit different then matter as we know it couldn’t form. If we were just a bit closer to the Sun we couldn’t survive and so on.

These things are all true. But in a Universe with inappropriate physical constants there will be no matter by definition, and certainly no sentient matter like us to notice.

If the Earth was closer to the Sun, homo sapiens would not exist as it currently does, but like as not some other species would, and it would also think how perfectly the world was formed for them. This concept, that we tend to think the Universe is so special to have created us, despite the fact that if not we wouldn’t be here to think about it, is often called the Anthropic Principle.

We believe that the improbability of life itself or us as a species is an example of the collision between billions of individually improbable events with the billions of events than could have occurred, and the billions of years in which they might have done so.

We end up having to think about such things with great care because our evolution on the African savannah didn’t really equip us to consider them. So we shouldn’t be surprised that we are surprised, but we still are.

â€œThis is rather as if you imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in â€” an interesting hole I find myself in â€” fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’ This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything’s going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.â€

Anyway, I slowly built an SVG file in Inkscape to reflect this family tree, but it was slow and rather a pain to maintain. So I’ve long wondered if it can be done better in LaTeX, with the Tikz graphics packages. Tikz is awesome, and while it has quite a learning curve as for LaTeX itself, I used it extensively in producing the slides for my recent inaugural lecture. This meant mixing the formulae with the graphics was pretty effortless. This is a slightly different problem.

A quick scan, showed that Tikz can be used to build flowcharts rather easily. So I began in this way, building nodes of people and then positioning them relative to each other. A bit tedious but much more pleasing that doing it in a vector graphics program, for me anyway. I rapidly ran into problems in building very large page styles. In particular odd things happened when I tried to include graphics (the image became the whole page, regardless of scaling).

I experimented briefly with the tikzposter package, but couldn’t really get anywhere with it – perhaps because I hadn’t started from the ground up with it. I think I looked at beamerposter but again ran into problems, possibly because I didn’t start that way. In the end it was the simple a0poster package that did it for me, anyway – although I had to force the font throughout to be “\tiny”.

I also discovered the hard way that the wrapfig package doesn’t seem to work within a Tikz node, so I’ve messily improvised by adding another node for the images of people and relating its position to the relevant biographical section.

I harvested some basic biographical information from Wikipedia and one other website.

This article contains links to materials and extra resources to my Inaugural Professorial Lecture, with the same name, delivered on 17th February 2016 at Ulster University.

Twitter

If you have any comments or questions, use the Twitter hashtag #nesssuff and I’ll pick them up later and try to address them. My Twitter ID is @ProfCTurner.

The Vote of Thanks will be given Sarah Flynn, whose Twitter ID is @sarahjaneflynn.

Synopsis

“Necessary and Sufficient: a look at elegance, efficiency and completeness in Engineering and its Mathematicsâ€

Engineers and Pure Mathematicians have a surprising amount in common, despite working at opposite ends of many problems; one at the totally theoretical end and the other at that of practical realisation, sometimes centuries apart. They both use tools created or designed mainly by other members of their own profession; they both enjoy testing things to destruction in order to explore how they work; and they both enjoy finding solutions to problems that cover all the requirements but which tend to do so in an efficient and elegant way.

This lecture explores how basic concepts that began with natural numbers to count livestock in antiquity eventually gave rise to complex numbers, and how techniques to measure buildings and the movement of the stars evolved into techniques to analyse data in totally new ways.

Some modern applications, ranging from every day examples such as photographs taken by smart-phones through to research applications, will also be considered.

Finally, the lecture will examine the implications for how Engineers can be educated to bring the power of some of humanityâ€™s most beautiful abstract ideas to bear on the practical problems that surround us in everyday life.

Lecture Slides

(“Director’s cut” and “Commentary/Video” to be uploaded at a later date).

– Videos not embedded, no pauses (~3 MB)

– Full Size Slides with pauses and embedded Video (~52 MB)

For those interested the slides were produced with PDFLaTeX, Beamer and Tikz. Diagrams with plots and positions of complex numbers are all calculated as the PDF is compiled. The presentation was stored in a git repository and a Makefile was used to produce the various versions.

A GitHub repository with some files missing (due to them being University property) is available here. But this does contain all Tikz diagram source code, cow images, and a LaTeX Beamer template aligned to the Faculty template that was produced. Faculty colleagues can request the required University images for their own presentations. The Makefile shows how to create different versions of the talk, with embedded or linked videos, and with or without pauses.

The content of the talk is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

I answered a question on Quora about the consequences of Newton’s law of Universal Gravitation being in inverse law and not an inverse square law for 60 seconds. The outcome is not so good, what with all matter acting as a black hole and all.

According to the Mirror, and I realise this may not be the strongest start,

“But bookies William Hill said the odds of coincidentally buying a car with that reg to take to Argentina were about 13 million to one.”

Unfortunately the bookies’ article isn’t directly referenced so far as I can see, nor is the working out shown, but I’m instantly suspicious of such statements – a lot of assumptions (and hindsight) may have gone into this. If my Google-Fu is weak and you spot the direct link, please let me know.

Note that the BBC team strongly denies that the plate was bought specially.

To me, this is a classic potential case of underestimating how numbers stack up. Coincidences happen all the time. There are many plates that people can take offence at, many people buying them, many people travelling to somewhere it may have a different meaning and so on.

But in any case, the 13 million to one odds are presumably presented as a tacit statement that it couldn’t have been a coincidence.

The UK lottery uses 49 balls, of which you have to guess 6 correctly. The number of ways of choosing 6 from 49 is given by a relatively simple formula.

Let’s work that out

That turns out to be 13,983,816.

So, from a naive point of view the odds of winning the lottery are almost 14 million to 1. Yes, that is very unlikely. Famously much more unlikely than lots of other events, but it happens all the time.

“Unlikely” events are more psychologically available, we notice and remember them more, it can mean we assign significance to them which isn’t deserved.

If Jeremy Clarkson, or one of the others in the team had won the lottery that would have made the news too, but it wouldn’t imply a stitch up.

Yesterday I was finally nominated for the Ice Bucket Challenge, I had actually thought this inevitable for the reasons in this post, but then it all kind of passed by. As it happens, it was my Daughter Aimee’s fault in the end. :-).

So I thought I would use the opportunity to bore people about the arithmetic behind it, in the end, I suspect that wasn’t too coherent since I was being watered at the time by my older Daughter Aimee, while my younger Daughter Matilda generally screamed at both of us.

So here is what I was trying to say. I like to talk to my students about a new version of an old problem. There’s an old story about a chess game where the winner will take away a certain amount of rice. The amount is calculated by having one grain on the first square, two on the next, four on the next, and so on, doubling all the way to the 64th square. My modified version is to consider coins, a UK 10 pence piece to be precise, piling up on the squares as we go along. The question is, how high is the pile of coins on the last square? I encourage you to guess, approximately what sort of size that is.

We double each time, so the number of coins on the first eight squares are 1, 2, 4, 8, 16, 32, 64, 128.
Another way to write that is20, 21, 22, 23, 24, 25, 26, 27.

So, we are going along 64 squares, and start at 0, so the number of coins on the last square will be263. This is a big number, but just how big? Do we change our guess on this information?

Now 263 = 9,223,000,000,000,000,000 = 9.223 x 1018
and a ten pence piece is 1.85 mm thick, or 0.00185 m, so when we multiply these we get

17,063,000,000,000,000 m = 1.7063 x1016 m.

Wow. That seems like a lot. Just how big is that number as a distance?

To put it in perspective…

by the 19th square, the coins are higher than the radio mast on the Empire State building;

by the 29th square, the pile of coins would have reached the Moon;

by the 48th square, the pile of coins would have reached the Sun;

by the 63rd square, one from the end, the coins are a light year high.

This is an example of the staggering power of exponential growth, and just how unintuitive it is.

So what has this got to do with the Ice Bucket Challenge? Well suppose you start with a single individual, who then nominates three people, each of whom nominate three people. Now on the Chess Board we have 1, 3, 9, 27 and so on. This is, again, exponential growth with powers of three rather than two (actually somewhat higher growth).

Mathematically these are often called Geometric Progressions or just G.P.s for short. These are sequences of the type

a, ar, ar2, ar3, ar4, …

There is a formula that can be derived (it’s not hard, the derivation is on the above Wikipedia link) for the Sum of the first terms. (In Mathematics, contrary to popular opinion, a Sum specifically means the result of an addition process).

In the above case we can see that a = 1 because that’s the first number in 1, 3, 9, … and r = 3 because that is the number we are multiplying by each time. So here

In other words, every “generation” each person nominates a further three people, so the number of people added each generation ramps up exponentially, and the total number involved increased rapidly too.

Generation

New People

Cumulative Total

1

1

1

2

3

4

3

9

13

4

27

40

5

81

121

6

243

364

7

729

1,093

8

2,187

3,280

9

6,561

9,841

10

19,683

29,524

11

59,049

88,573

12

177,147

265,720

13

531,441

797,161

14

1,594,323

2,391,484

15

4,782,969

7,174,453

16

14,348,907

21,523,360

17

43,046,721

64,570,081

18

129,140,163

193,710,244

19

387,420,489

581,130,733

20

1,162,261,467

1,743,392,200

21

3,486,784,401

5,230,176,601

22

10,460,353,203

15,690,529,804

So you can see that after just 22 steps in the nomination process, you have literally more people than you have on the planet. Pyramid selling schemes become untenable for this reason too… you quickly run out of suckers on any given land mass.

So if this really ran its course unimpeded (and I know that the original challenge was money or dowsing, with a requirement to undertake the challenge in 48 hours (some say 24) then the whole planet would have done the challenge in at most six weeks. If each person had used say 4 litres of water then at the time of writing the 7,259,289,122 people alive today would have collectively used 29 billion litres of water. Since an Olympic swimming pool has 2.5 million litres of water, that represents over 11,614 swimming pools filled with water. While it is easy to criticise those who have been uneasy about the water wastage here, to be honest they have a good point. Also one organisation would have billions of charity income rather than it being spread.

Clearly this has not happened[citation needed] so people have run out on enthusiasm here and there. Anyway, have fun, and give some money to something you love, if you want, when you want to.

Today, two of my friends independently sent me a story about Tau Day which I had hitherto never heard of. One of them asked for me comment about whether this had any point to it. At first I thought the article was just mathematical trolling, thought about it a bit more, thought there might be a real point to it, thought some more and concluded it seemed rather silly.

The argument is about whether the mathematical constant pi, would be better being replaced throughout mathematics with an alternative tau, which is just twice pi (in other words, replacing pi everywhere with a half of this tau). It’s suggested that formulae with tau will be more simple.

Basic Geometry

So this is all about the fact that pi was defined historically as the ratio of the circumference to the diameter of the circle, a very old classical reference stemming back to Greek geometry (incidentally pi is also known as Archimedes’ constant since he attempted to calculate an approximation to it). Once upon a time, the formula used in schools would have been:

related the circumference C to the diameter D. But generally now, we use the radius r rather than the diameter. And so that gives us (for circumference and area):

The argument for tau begins by observing the extra 2 in the first formula, and wouldn’t be nicer if we just defined tau to be twice pi so that these formula would be so much nicer. Would they?

Set aside for the moment the fact that pi is probably the most recognisable Greek letter in the world that speaks languages based on the Latin alphabet (aside from those that are, or appear to be the same). Set aside the fact that tau is used for other specific purposes in much of modern Mathematics, and in particular in the discipline of Topology. The first formula might be nicer, but the second one is probably worse, and by enough to make the improvement of the first rather parlous. OK. But the article talks about this being the problem behind radians, so maybe that’s where we get the big gain. Let’s explore that.

Radians instead of Degrees

There’s nothing particularly clever about using degrees. It’s an arbitrary choice (360 degrees in a circle) that probably owes a lot to do with historical factors in one civilisation. It is true that when you start to do some significant mathematics with degrees, it starts to look quite unwieldy. The classic two formulae to consider are the length of an arc and area of a sector.

Suppose we have a circle of radius r and we want to work out the length of an arc (a part of the circumference) where the angle subtending this arc is theta degrees (don’t panic, no more Greek to come). Then in degrees the formula will be:

The reason why is that the fraction on the left is the fraction of the relevant angle out of all the angle available, multiplied by the total arc length available (the whole circumference). The formula is not beautiful, and the similar formula for sector area is also a big ugly.

You will note that in both cases there is a 360 on the bottom of the fraction and a 2 pi on the top. This looks like nature’s way of trying to tell us something. What would happen if we used an unit of angle so that, instead of having 360 of them in a circle, we had 2 pi of them in a circle (proponents of tau will just say tau of them in a circle)? The formula, derived using the same logic, become much nicer.

So we get

Now these are beautiful, elegant formulae, and the underpinning of why radians (the unit of angle we are talking about here) are used instead of degrees in much of higher mathematics, the formula are much simpler (particularly true when using calculus). Also, look at that first formula, it has all the resonance of F = ma. 1 unit of arc length is found in a circle of radius 1 unit with an angle of 1 radian. So beautiful is this that it used as the definition of the radian in many books. So far, so good. Did we really need tau to produce these? Does it matter that it it tau and not pi that cancels out? I can’t see why.

Fourier Series

Another example owes to the work of Fourier, who showed that repeating patterns can be broken into sums of the most basic repeating functions, the ones that are most simple are the sine and cosine functions. These are used to model waves of any sort which are of course ubiquitous in nature. It turns out you can build up more odd shapes like triangular and square waves out of these sinusoidal ones. Being able to do this is important in many aspects of Science and Engineering, and eventually this theory leads to all sorts of cool stuff like the way data is compressed in photos and more.

This graph is labelled in degrees, so you can imagine if we change it to radians as we should then the 360 would become 2 pi. This is where again, proponents of tau will argue that replacing the 360 with a simple tau makes things easier. And it does, if all you want to do is to label that diagram. But the foundation of Fourier theory is building functions up in combinations of these:

The formulae you need to be able to deal with to do this are (among others):

which I grant, strike fear into the hearts of many. But they don’t look nicer with tau (and are a little more awkward):

Remember, the people arguing for tau are claiming it simplifies formulae, not making them look worse.

Euler’s Identity

Finally, I cannot leave this without talking about Euler’s identity considered by most mathematicians (including myself) to be one of the most beautiful results in Mathematics.

This result can be written in a few ways, but this way is very commonly used. This is because in this form you can see how this identity connects the five most important numbers of Mathematics: 0, 1, pi, i and e. With tau, it just doesn’t have the same beauty:

How interesting, today I learned that Derek Burgess, the PhD supervisor of my PhD supervisor (Brian McMaster) was himself supervised by Frank Smithies. With a little help from the Mathematical Genealogy Project this has helped me work out my academic “parentage”.

A few days later, this interesting article showed the results of professional physicists on just how little such little black holes could grow, in some cases even if their lifetime was not restricted. Enjoy.